20110207, 17:56  #1 
I quite division it
"Chris"
Feb 2005
England
4035_{8} Posts 
Primes from powers of 2 strings.
Create primes by arranging the strings produced from progressive powers of 2 from 2^0 to 2^n. i.e.
A prime using the string "1", none. A prime using the strings "2" and"1", none. A prime using the strings "4", "2" and "1", 421 or 241. etc. No taking parts of strings. All strings from 1 to your chosen n must be used. What is the largest prime you can make? The largest prime for the strings in order or near order, near reverse etc? The largest prime that is also a twin (+2 or 2) or has other interesting properties? Kudos for posting small, efficient code. If this is known or on OEIS create a similar more interesting puzzle! 
20110207, 18:50  #2 
Mar 2007
Austria
302_{10} Posts 
How about this small little PARIGP function:
Code:
for(z=1,100,y=2^z;for(x=1,z1,y*=10^(ceil(log(2)*(zx)/log(10)));y+=2^(zx));y*=10;y++;write("cand_puzzle.txt",y)) z(1)=21 z(2)=421 z(3)=8421... upto z=100.(you can ofcourse go further if you want) Then pump the output into PFGW and hopefully I'll find something! 
20110207, 19:10  #3  
1976 Toyota Corona years forever!
"Wayne"
Nov 2006
Saskatchewan, Canada
4,373 Posts 
Quote:
i.e. in 1,2,4,8,16,32  the 16 and the 32 can't be split as in 41823621? It "appears" that every second string starting with 1,2 add up to a multiple of 3 and so CANNOT be prime. 

20110207, 20:25  #4 
Jun 2003
13·19^{2} Posts 
Basic restrictions  all powers of two end in an even digit except 1. So only permutations ending in 1 has a chance of being prime. Also, for even n, the sum of digits of 2^0 .. 2^n is divisible by 3. So only odd 'n' has a chance of yielding primes.

20110207, 20:56  #5 
I quite division it
"Chris"
Feb 2005
England
31×67 Posts 

20110208, 11:26  #6 
Mar 2010
On front of my laptop
7×17 Posts 
220 digits
Code:
8589934592838860881928687194767366710886465536645368709125242885124294967296419430440964343597383683355443232768322684354562621442562147483648209715220482171798691841677721616384161342177281310721048576107374182410241281 8589934592_8388608_8192_8_68719476736_67108864_65536_64_536870912_524288_512_4294967296_4194304_4096_4_34359738368_33554432_32768_32_268435456_262144_256_2147483648_2097152_2048_2_17179869184_16777216_16384_16_134217728_131072_1048576_1073741824_1024_128_1 
20110208, 15:28  #7 
I quite division it
"Chris"
Feb 2005
England
100000011101_{2} Posts 
Very nice. How did you find it?

20110208, 16:17  #8  
Mar 2007
Austria
2×151 Posts 
Quote:
Will try another orientation and hopefully I can beat the 220 digit record! 

20110208, 19:49  #9 
Aug 2006
13437_{8} Posts 
I have a 1224digit prime using the powers of 2^0 to 2^88:
Code:
309485009821345068724781056154742504910672534362390528773712524553362671811952643868562622766813359059763219342813113834066795298816967140655691703339764940848357032784585166988247042417851639229258349412352120892581961462917470617660446290980731458735308830223145490365729367654415111572745182864683827275557863725914323419136377789318629571617095681888946593147858085478494447329657392904273924722366482869645213696236118324143482260684811805916207174113034245902958103587056517122951479051793528258561475739525896764129287378697629483820646436893488147419103232184467440737095516169223372036854775808461168601842738790423058430092136939521152921504606846976576460752303423488288230376151711744144115188075855872720575940379279363602879701896396818014398509481984900719925474099245035996273704962251799813685248112589990684262456294995342131228147497671065614073748835532870368744177664351843720888321759218604441687960930222084398046511104219902325555210995116277765497558138882748779069441374389534726871947673634359738368171798691848589934592429496729621474836481073741824536870912268435456134217728671088643355443216777216838860841943042097152104857652428826214413107265536327681638481924096204810245122561281632864421 
20110208, 23:55  #10  
Nov 2003
7460_{10} Posts 
Quote:
Using your rules, I expect only finitely many; the numbers grow too fast. 

20110209, 00:01  #11 
I quite division it
"Chris"
Feb 2005
England
4035_{8} Posts 

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